Bayesian Incentive Compatibility

• Bayesian Incentive Compatibility is the principle where agents maximize expected utility by truthfully reporting private information based on a common prior.
• The concept underlines the trade-offs between efficiency, fairness, and incentive constraints in mechanism design, contrasting OBIC with robust LROBIC approaches.
• Robust Bayesian Incentive Compatibility (LROBIC) ultimately aligns with ex post strategy-proofness, reintroducing classical impossibility results in mechanism design.

Bayesian Incentive Compatibility (BIC) is a foundational solution concept in mechanism design. It requires that every agent maximizes expected utility by reporting their private information (type) truthfully, where the expectation is taken over other agents’ types according to a commonly known prior. For researchers, BIC is especially central in domains where ex post incentive constraints (dominant strategy incentive compatibility; strategy-proofness) conflict with desirable efficiency or fairness properties, or where only prior-based reasoning is tractable.

1. Formal Definition and Stochastic Dominance in BIC

In the random assignment model, each agent $i$ possesses a strict ranking $P_i$ over $n$ objects. An assignment mechanism $Q$ maps the profile of preferences $P = (P_1, ..., P_n)$ to a bistochastic matrix encoding the probability shares allocated to each agent-object pair.

Strategy-proofness requires that for any possible realization of the other agents’ preferences $P_{-i}$, the assignment resulting from truthful reporting ($Q_i(P_i, P_{-i})$) stochastically dominates, according to first-order stochastic dominance (FOSD), the assignment resulting from any misreport $\tilde{P}_i$.

Bayesian incentive compatibility (specifically, ordinal Bayesian incentive compatibility, OBIC, in this context), is weaker: $$Q \text{ is OBIC with respect to a prior } p \iff \forall i,\,\forall P_i,\tilde{P}_i:\quad q_i(P_i) \succcurlyeq_{P_i} q_i(\tilde{P}_i)$$ where

$$q_i(P_i) = \sum_{P_{-i}} p(P_{-i}) Q_i(P_i, P_{-i})$$

is the interim share vector (expected shares over $P_{-i}$), and $\succcurlyeq_{P_i}$ denotes FOSD with respect to $P_i$.

In OBIC, truth-telling is optimal in terms of the stochastically maximal expected share, but only in expectation over others’ preferences, not for every realization. This relaxation allows a larger class of mechanisms to qualify as incentive compatible under the Bayesian (prior-dependent) notion.

2. Structure of OBIC Under Uniform Priors

For the uniform prior, in which all preference profiles are equally likely, the paper establishes a broad positive result:

• Theorem 1: Any mechanism $Q$ that is neutral (objects treated symmetrically with respect to agent reports) and satisfies elementary monotonicity is OBIC with respect to the uniform prior (U-OBIC).

Elementary monotonicity is a local property: if $P_i$ and $\tilde{P}_i$ differ by adjacent swap of objects $a$ and $b$, then

$$Q_{i, b}(P_i, P_{-i}) \geq Q_{i, b}(\tilde{P}_i, P_{-i})\,,\quad Q_{i, a}(P_i, P_{-i}) \leq Q_{i, a}(\tilde{P}_i, P_{-i})$$

for all $P_{-i}$. This ensures that moving an object up in the ranking can only increase its allocated share and moving it down can only decrease it.

Neutrality requires allocations to be invariant under object permutations: for any permutation $\sigma$,

$Q_{i,a}(P) = Q_{i,\sigma(a)}(P^\sigma)$

where $P^\sigma$ is the profile with permuted objects.

The implication is that all simultaneous eating algorithms, including the probabilistic serial (PS) mechanism, are U-OBIC. These mechanisms are not ex post strategy-proof, but they are Bayesian strategy-proof under the uniform prior, vastly enlarging the mechanism class achievable relative to ex post constraints.

3. Locally Robust OBIC, Robustness, and Equivalence to Strategy-Proofness

The paper introduces locally robust OBIC (LROBIC), requiring that incentive compatibility hold not only for a specific prior but for all i.i.d. priors within an open neighborhood (i.e., all $p'$ such that $|p(P) - p'(P)| < \epsilon$ for all $P$ for some $\epsilon > 0$).

Theorem 2 (Equivalence Theorem):

If $Q$ is LROBIC and satisfies elementary monotonicity, then $Q$ is strategy-proof (i.e., ex post IC). Conversely, any strategy-proof mechanism is LROBIC.

The key consequence is that LROBIC with mild monotonicity property collapses to full ex post incentive constraints. Any attempt to robustify Bayesian incentive compatibility in this sense brings back the impossibility boundaries known for strategy-proofness.

4. Efficiency, Fairness, and Strong Impossibility Theorems

Ordinal efficiency demands that no other assignment stochastically dominates the mechanism’s output for all agents, with at least one agent strictly better off. Equal treatment of equals requires agents with identical preferences to receive identical (probabilistic) allocations.

Bogomolnaia and Moulin (2001) established that, for $n \geq 4$, it is impossible to achieve (i) strategy-proofness, (ii) ordinal efficiency, and (iii) equal treatment of equals simultaneously.

The paper’s main strengthening:

For $n \geq 4$, no mechanism is simultaneously LROBIC, ordinally efficient, and treats equals equally. That is, even relaxing from ex post to a robust Bayesian version of incentive compatibility (LROBIC), if efficiency and fairness are required, the impossibility frontier remains unchanged.

Formally,

$$\text{No }Q\text{ is LROBIC, ordinally efficient, and equal treatment of equals for }n \geq 4$$

Corollary: For $n \geq 4$, popular Bayesian mechanisms like PS that are U-OBIC cannot be made robust (LROBIC) if efficiency and fairness properties are desired.

5. Mechanism Design Implications and Summary Table

The paper demonstrates a critical hierarchy:

Property	Uniform Prior OBIC (U-OBIC)	LROBIC
Definition	OBIC under uniform prior	OBIC under all nearby i.i.d. priors
Characterization	Neutral + elem. monotonicity ⇒ U-OBIC	(+) ⇒ strategy-proof
Example Mechanism	PS mechanism is U-OBIC	PS is not LROBIC
Impossibility	No impossibility: PS is U-OBIC, efficient, fair	LROBIC + efficiency + equality impossible $(n\geq4)$

U-OBIC, under neutral priors and elementary monotonicity, allows for broad classes of mechanisms combining efficiency and fairness with Bayesian incentive compatibility. In contrast, robustness (LROBIC) collapses to ex post IC and carries over classic impossibility theorems without expanding the feasible set of mechanisms.

6. Technical Summary and Formal Results

Key mathematical formulations:

• OBIC Condition:

$$q_i(P_i) = \sum_{P_{-i}} p(P_{-i}) Q_i(P_i, P_{-i})\,,\qquad Q\text{ is OBIC if } q_i(P_i) \succcurlyeq_{P_i} q_i(\tilde{P}_i)$$

(FOSD on interim shares given own preference vs. misreport)

• Elementary Monotonicity (for adjacent swap $a \leftrightarrow b$):

$$Q_{i, b}(P_i, P_{-i}) \geq Q_{i, b}(\tilde{P}_i, P_{-i}) \,,\quad Q_{i, a}(P_i, P_{-i}) \leq Q_{i, a}(\tilde{P}_i, P_{-i})$$

• Robustness/Equivalence:

$$Q \text{ is LROBIC and elem. monotonic} \iff Q\text{ is strategy-proof}$$

• Strengthened Impossibility:

$$n \geq 4\quad \implies \quad \nexists Q\text{ LROBIC, ordinally efficient, and equal treatment of equals}$$

7. Broader Perspective and Ongoing Challenges

The separation between weak OBIC (prior-specific) and robust OBIC (LROBIC) highlights that the practicable power of Bayesian IC is sensitive to assumptions about agents’ belief structures and the symmetry (neutrality) of priors. For the random assignment problem, uniform-prior OBIC mechanisms like PS are justifiable as soon as belief structures are precisely specified and symmetric, but become untenable under mild robustness requirements on agents’ beliefs.

This analysis brings previously “positive” Bayesian-IC mechanism results into sharper focus, clarifies boundary conditions for impossibility, and exposes the potential fragility of incentive compatibility when robust prior-independence or non-manipulability to local perturbations is sought. The equivalence with ex post IC for robust Bayesian IC settings (with mild monotonicity) is especially significant, as it implies that classic negative results reassert themselves even under a modest strengthening of Bayesian IC.

References:

• Sulagna Dasgupta and Debasis Mishra ("Ordinal Bayesian incentive compatibility in random assignment model" (Dasgupta et al., 2020))
• Bogomolnaia and Moulin (2001): "A new solution to the random assignment problem"
• Mennle and Seuken (2021): "Partial strategyproofness: Relaxing strategyproofness for the random assignment problem"